We have a segment AC, with the point B lying between A and C.
The ratio AB to BC is 5:6.
The coordinates for A and C are:
A=(2,-6)
C=(-4,2)
We can calculate the coordinates of B for each axis, using the ratio of 5:6.
[tex]\begin{gathered} \frac{x_a-x_b}{x_b-x_c}=\frac{2-x_b}{x_b+4}=\frac{5}{6}_{} \\ 6\cdot(2-x_b)=5\cdot(x_b+4) \\ 12-6x_b=5x_b+20 \\ -6x_b-5x_b=20-12_{} \\ -11x_b=8 \\ x_b=-\frac{8}{11}\approx-0.72\ldots \end{gathered}[/tex]
We can do the same for the y-coordinates:
[tex]\begin{gathered} \frac{y_a-y_b}{y_b-y_c}=\frac{-6-y_b}{y_b-2}=\frac{5}{6} \\ 6(-6-y_b)=5(y_b-2) \\ -36-6y_b=5y_b-10 \\ -6y_b-5y_b=-10+36 \\ -11y_b=26 \\ y_b=-\frac{26}{11}\approx-2.36\ldots \end{gathered}[/tex]
The coordinates of B are (-8/11, -26/11).
